Set theory: difference between belong/contained and includes/subset?

Whenever you come across something like this and it trips you up, you might want to look at particular examples. For instance, consider $\{4\}$. $\{4\} \subset \{4\}$, but $\{4\} \in \{4\}$ is false, since the only member of $\{4\}$ is $4$, not $\{4\}$. It may have tripped you up that "includes" and "contains" in everyday language usually qualify as synonyms. They don't here, and the terms get defined by the definitions for $\in$ and $\subset$. You might want to prove that $A \subset A$ for any set $A$ as it can get proven in a line or two.

The point is that every set is a subset of itself, namely $A\subseteq A$ - always.

However $\in$ does not have this property, for example $\varnothing$ has no elements, in particular $\varnothing\notin\varnothing$. However $\varnothing\subseteq\varnothing$.

To make matters worse, $\varnothing\in\{\varnothing\}$ as well $\varnothing\subseteq\{\varnothing\}$. However $\varnothing\neq\{\varnothing\}$ since the empty set has no elements and the singleton $\{\varnothing\}$ has an element.

In the axiomatic approach to set theory, the commonly used axioms of ZF dictate that $A\notin A$ for every set $A$. This is a result of something called the axiom of regularity, or axiom of foundation. However there are useful instances of non-well founded set theory in which some sets have the form $x=\{x\}$. For more information on that: When is $x=\{ x\}$?

Regardless to that, it is always the case that $A\subseteq A$, and always the case that for some $B$ we have $B\notin B$.